In weightlifting, the term "going to failure" means moving the weight through the normal range of motion, using good form, until you cannot complete the last rep. So let's say you're doing barbell curls. You can't quite do ten reps at the weight you're using. When you get to the tenth one, you curl it but you can't lift it all the way up without changing your form. So you bring it up as far as you can while maintaining good form. You have "gone to failure."
The main alternative to going to failure is to have a predetermined number of sets and reps you will do. For example, 3 sets of 8 reps. Or one set of 9, one set of 8, and one set of 7 reps.